Some Evaluations About Coefficients Boundaries for Specific Classes of Bi-Univalent Functions

Abstract

New subclasses of bi-univalent functions with bounded boundary rotation are presented in this study. We acquired estimates for the initial coefficients $\left|a_2\right|$, $\left|a_3\right|$ and $\left|a_4\right|$. Furthermore, we have verified the specific situations satisfying the famous hypothesis of Brannan and Clunie. Additionally, we have obtained the well-known Fekete–Szegö inequality for the newly identified bi-univalent function subclasses. Our results not only improve, but also extend several existing results as particular cases.

1. Introduction

In the complex plane, let $\mathbb{D}$ refer to the open unit disc so that

$$\mathbb{D}=\left\{\xi \in C:\left|\xi \right|<1\right\}$$

Also, let $\mathbb{A}$ be the class of all functions $j:\mathbb{D}⟶C$ with the following representation:

$$j\left(\xi \right)=\xi +\sum _{n⩾2}{a}_n{\xi }^n,\left(n\in \mathbb{N};\xi \in \mathbb{D}\right).$$

(1)

Under the criteria that $j\left(0\right)=j^{\prime }\left(0\right)-1=0$, these functions are normalized; also, they are analytic on $\mathbb{D}$. A subclass of $\mathbb{A}$ is characterized by the symbol $\mathbb{S}$, which consists of the univalent functions in $\mathbb{D}$. Additionally, recalling ${\mathbb{S}}^{*}\left(\beta \right)$ and $\mathbb{C}\left(\beta \right)$, the starlike and convex functions of order $\beta$ are defined as follows:

$${\mathbb{S}}^{*}\left(\beta \right)=\left\{j\in \mathbb{S}:Re\left({\displaystyle \frac{\xi j^{\prime }\left(\xi \right)}{j\left(\xi \right)}}\right)>\beta ,0\le \beta <1\right\},$$

(2)

$$\mathbb{C}\left(\beta \right)=\left\{j\in \mathbb{S}:Re\left(1+{\displaystyle \frac{\xi j^{\prime \prime }\left(\xi \right)}{j^{\prime }\left(\xi \right)}}\right)>\beta ,0\le \beta <1\right\}.$$

(3)

Keep in consideration that $\mathbb{C}\left(0\right)=\mathbb{C}$ and ${\mathbb{S}}^{*}\left(0\right)={\mathbb{S}}^{*}$ refer to convex and starlike functions. Robertson [1] first presented the classes ${\mathbb{S}}^{*}\left(\beta \right)$ and $\mathbb{C}\left(\beta \right)$, which MacGregor later examined in [2], Schild [3], Pinchuk [4], and Jack [5]. Moreover, a function $j\in \mathbb{S}$ given in (1) with $j^{\prime }\left(\xi \right)\ne 0$ is categorized as belonging to the close-to-convex function of order $\beta$, denoted as $\mathbb{K}\left(\beta \right),$ if and only if there exists a convex function g such that (see Kaplan [6])

$$Re\left({\displaystyle \frac{j^{\prime }\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right)>\beta ,\left(0\le \beta <1,\xi \in \mathbb{D}\right).$$

(4)

An equivalent formulation would involve the existence of a starlike function g such that

$$Re\left\{{\displaystyle \frac{\xi j^{\prime }\left(\xi \right)}{g\left(\xi \right)}}\right\}>\beta ,\left(0\le \beta <1,\xi \in \mathbb{D}\right).$$

(5)

Note that $\mathbb{K}\left(0\right)=\mathbb{K}$ and

$$\mathbb{C}\subseteq {\mathbb{S}}^{*}\subseteq \mathbb{K}\subseteq \mathbb{S}.$$

(6)

Following [7], a function $j\in \mathbb{S}$ is called quasi-convex and of order Beta if and only if

$$Re\left\{{\displaystyle \frac{{\left(\xi j^{\prime }\left(\xi \right)\right)}^{\prime }}{g^{\prime }\left(\xi \right)}}\right\}>\beta ,\left(0\le \beta <1,\xi \in \mathbb{D},g\in \mathbb{C}\right).$$

(7)

We will use the symbol $Q\mathbb{C}\left(\beta \right)$ for the family of these types of functions.

From (5) and (7), it follows that

$$j\in Q\mathbb{C}\left(\beta \right)\iff \tilde{j}\in \mathbb{K}\left(\beta \right),\tilde{j}\left(\xi \right)=\xi j^{\prime }\left(\xi \right).$$

(8)

Note that $Q\mathbb{C}\left(0\right)=Q\mathbb{C}.$ If the range of a function has a bounded boundary rotation, then the function is said to have bounded boundary rotation if it is analytic and locally univalent inside a given simply connected domain. The entire variation of the direction angle of the tangent to the boundary curve during a full circuit is referred to as bounded boundary rotation. The function then maps the domain, denoted as $\mathbb{D}$, conformally onto an image domain with maximum boundary rotation $k\pi$.

Let us start with following definitions.

Definition 1

([8]). Let ${\mathbb{P}}_k\left(\beta \right)$ denote the family analytic functions p on $\mathbb{D}$, with conditions $p\left(0\right)=1$ and

$${\int }_0^{2\pi }\left|{\displaystyle \frac{Re\left(p\left(\xi \right)\right)-\beta }{1-\beta }}\right|d\theta \le k\pi .$$

(9)

where $\xi =r{e}^{i\theta },$ $k\ge 2,$ $0\le \beta <1.$

We see that

(i)

${\mathbb{P}}_k\left(0\right)={\mathbb{P}}_k(k\ge 2)$ is the set of functions with the real portion constrained in the mean on $\mathbb{D}$ [9,10].

(ii)

${\mathbb{P}}_2\left(\beta \right)=\mathbb{P}\left(\beta \right)$ is the subset of positive real part functions of order $\beta .$

(iii)

${\mathbb{P}}_2\left(0\right)\equiv \mathbb{P}$ is the subclass of Caratheodory functions [11].

Let ${\mathbb{S}}_k\left(\beta \right)$ be the set of analytic functions j in $\mathbb{D}$ with $j\left(0\right)=j^{\prime }\left(0\right)-1=0$, with

$${\displaystyle \frac{\xi j^{\prime }\left(\xi \right)}{j\left(\xi \right)}}\in {\mathbb{P}}_k\left(\beta \right),0\le \beta <1.$$

${\mathbb{S}}_k\left(\beta \right)$ generalizes ${\mathbb{S}}^{*}\left(\beta \right)$, which was given by Robertson [1]. When $\beta =0$, then ${\mathbb{S}}_k\left(0\right)\equiv {\mathbb{S}}_k$, which represents all bounded radius rotation functions. Consequently, the functions $j\in {\mathbb{S}}_k$ can be expressed as

$$j\left(\xi \right)=\xi exp\left({\int }_0^{2\pi }-log\left(1-\xi e^{it}\right)d\lambda \left(t\right)\right),$$

such that

$${\int }_0^{2\pi }d\lambda \left(t\right)=2\mathrm{and}{\int }_0^{2\pi }\left|d\lambda \left(t\right)\right|\le k,k\ge 2.$$

(10)

Let ${\mathbb{C}}_k\left(\beta \right)$ refer to the class of functions $j\left(\xi \right)$ on $\mathbb{D}$, satisfying $j\left(0\right)=j^{\prime }\left(0\right)-1=0$ and

$$1+{\displaystyle \frac{\xi j^{\prime \prime }\left(\xi \right)}{j^{\prime }\left(\xi \right)}}\in {\mathbb{P}}_k\left(\beta \right),0\le \beta <1.$$

If $\beta =0,$ then ${\mathbb{C}}_k\left(0\right)\equiv {\mathbb{C}}_k$. The set of bounded boundary rotation functions ${\mathbb{C}}_k$ was investigated by Paatero [12]. Hence, the functions $j\in {\mathbb{C}}_k$ can be represented as

$$j^{\prime }\left(\xi \right)=exp\left({\int }_0^{2\pi }-log\left(1-\xi e^{it}\right)d\lambda \left(t\right)\right),$$

where $\lambda$ is determined by (10). The class ${\mathbb{C}}_k\left(\beta \right)$ generalizes the class of convex functions $\mathbb{C}\left(\beta \right)$ of order $\beta$ introduced by Robertson [1]. Pinchuk [9] gave the relations below:

$$j\left(\xi \right)\in {\mathbb{C}}_k\left(\beta \right)\iff 1+{\displaystyle \frac{\xi j^{\prime \prime }\left(\xi \right)}{j^{\prime }\left(\xi \right)}}\in {\mathbb{P}}_k\left(\beta \right),$$

$$j\left(\xi \right)\in {\mathbb{S}}_k\left(\beta \right)\iff {\displaystyle \frac{\xi j^{\prime }\left(\xi \right)}{j\left(\xi \right)}}\in {\mathbb{P}}_k\left(\beta \right),$$

and

$$j\left(\xi \right)\in {\mathbb{C}}_k\left(\beta \right)\iff \xi j^{\prime }\left(\xi \right)\in {\mathbb{S}}_k\left(\beta \right).$$

Pinchuk [9] also established that functions in ${\mathbb{C}}_k$ are close-to-convex in $\mathbb{D}$ when $2\le k\le 4$ and therefore, they are univalent. Brannan [13] demonstrated that ${\mathbb{C}}_k$ is a subclass of the class $\mathbb{K}\left(\beta \right)$ of functions that are close-to-convex of order $\beta ={\displaystyle \frac{k}{2}}-1.$ For $j\left(\xi \right)\in {\mathbb{C}}_k\left(\beta \right)$ and $k=2,3,$ the following sharp results hold: $\left|a_2\right|\le {\displaystyle \frac{k}{2}}$ and $\left|a_3\right|\le {\displaystyle \frac{k^2+2}{6}}$ (refer to [14] for details). The Koebe one-quarter theorem (see [15]) guarantees that the image of $\mathbb{D}$ under any univalent function $j\in S$ contains a disk centered at the origin with a radius of $1/4$. Hence, every function $j\in S$ possesses an inverse function $j^{-1}:$$j\left(\mathbb{D}\right)\to \mathbb{D},$ which satisfies the conditions

$$j^{-1}j\left(\xi \right)=\xi \left(\xi \in \mathbb{D}\right),$$

and

$$j\left(j^{-1}\left(w\right)\right)=w\left(\left|w\right|<r_0\left(j\right),r_0\left(j\right)\ge {\displaystyle \frac{1}{4}}\right).$$

The inverse function $j^{-1}$ is defined as

$$g\left(w\right)=j^{-1}\left(w\right)$$

$$=w-a_2{w}^2+\left(2a_2^2-a_3\right)w^3-(5a_2^3-5a_2{a}_3+a_4)w^4+\dots .$$

$$=w+{\displaystyle \sum _{n=2}^{\infty }}{A}_n{w}^n\left(n\in \mathbb{N}\right).$$

(11)

A function $j\in \mathbb{S}$ is called bi-univalent on $\mathbb{D}$ if both j and its inverse function $j^{-1}$ are univalent on $\mathbb{D}$, assuming that $\mathbb{D}\subseteq j\left(\mathbb{D}\right)$. Let $\Sigma$ be the set of these bi-univalent mappings. We see that

$${\displaystyle \frac{\xi }{1-\xi }},{\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+\xi }{1-\xi }}\right),-log(1-\xi )\in \Sigma .$$

But, the Koebe function ${\displaystyle \frac{\xi }{{\left(1-\xi \right)}^2}}\notin \Sigma .$ Similarly, ${\displaystyle \frac{\xi }{1-{\xi }^2}}$ and $\xi -{\displaystyle \frac{{\xi }^2}{2}}\notin \Sigma$ are two members from $\mathbb{S}$ (see [16]). Brannan and Taha [17] introduced bi-starlike functions of order $\beta$ denoted as ${\mathbb{S}}_{\Sigma }^{*}\left(\beta \right),$ and bi-convex functions of order $\beta$ denoted as ${\mathbb{C}}_{\Sigma }\left(\beta \right)$. The bi-univalent function classes gained a lot of attention and impetus, mostly as a result of Srivastava et al. [16]. Recently, several other researchers (see [18,19,20,21]) have presented and investigated a number of subclasses from $\Sigma$. They have derived bounds of the initial two coefficients in Taylor–Maclaurin series.

For

$$\varphi \left(\xi \right)=\xi +g_2{\xi }^2+g_3{\xi }^3+g_4{\xi }^4+\dots ,$$

(12)

we see that

$$\psi \left(w\right)=w-g_2{w}^2+\left(2g_2^2-g_3\right)w^3-(5g_2^3-5g_2{g}_3+g_4)w^4+\dots ,$$

(13)

where ${\varphi }^{-1}\left(\xi \right)=\psi \left(w\right).$

Definition 2

([22]). A function $j\in {\mathbb{A}}_{\Sigma },$ given in (1), is called a bi-close-to-convex function of order β if there exist bi-convex functions ϕ and ψ, with

$$Re\left({\displaystyle \frac{j^{\prime }\left(\xi \right)}{{\varphi }^{\prime }\left(\xi \right)}}\right)>\beta ,\left(0\le \beta <1\right),$$

and

$$Re\left({\displaystyle \frac{g^{\prime }\left(w\right)}{{\psi }^{\prime }\left(w\right)}}\right)>\beta \left(0\le \beta <1\right),$$

where g is given in Equation (11), and $\varphi \left(\xi \right)$ and $\psi \left(w\right)$ have the forms specified in Equations (12) and (13), respectively. The set of all bi-close-to-convex functions of order β is given the symbol ${\mathbb{K}}_{\Sigma }\left(\beta \right).$

Definition 3.

A function $j\in {\mathbb{A}}_{\Sigma },$ given by Equation (1), is called bi-quasi-convex function of order β if there exist bi-convex functions ϕ and ψ such that

$$Re\left({\displaystyle \frac{{\left(\xi j^{\prime }\left(\xi \right)\right)}^{\prime }}{{\varphi }^{\prime }\left(\xi \right)}}\right)>\beta \left(0\le \beta <1\right),$$

and

$$Re\left({\displaystyle \frac{{\left(w{g}^{\prime }\left(w\right)\right)}^{\prime }}{{\psi }^{\prime }\left(w\right)}}\right)>\beta \left(0\le \beta <1\right),$$

where g is defined by Equation (11), and $\varphi \left(\xi \right)$ and $\psi \left(w\right)$ have the forms specified in Equations (12) and (13), respectively. The set of bi-quasi-convex functions of order β is $Q_{\Sigma }\mathbb{C}\left(\beta \right).$

2. Preliminary Results

First, we state some definitions and lemmas which are essential for the coming investigations.

Lemma 1

([11,23]). If $\Phi \left(\xi \right)=1+{\displaystyle \sum _{n\ge 1}}{A}_n{\xi }^n\left(n\in \mathbb{N},\xi \in \mathbb{D}\right)\in {\mathbb{P}}_k\left(\beta \right),$ then

$$\left|A_n\right|\le k(1-\beta ),n\ge 1.$$

(14)

Consider $p,q\in {\mathbb{P}}_k\left(\beta \right)$, where

$$p\left(\xi \right)=1+\sum _{n\ge 2}{p}_n{\xi }^n\left(n\in \mathbb{N}\right),$$

(15)

and

$$q\left(\xi \right)=1+\sum _{n\ge 2}{q}_n{\xi }^n\left(n\in \mathbb{N}\right).$$

(16)

From Lemma 1, we have

$$\left|p_n\right|\le k(1-\beta ),\forall n\ge 1,$$

(17)

$$\left|q_n\right|\le k(1-\beta ),\forall n\ge 1.$$

(18)

Lemma 2

([24]). If $\Phi \left(\zeta \right)=\xi +{\displaystyle \sum _{n\ge 2}}{B}_n{\xi }^n\left(n\in \mathbb{N},\xi \in \mathbb{D}\right)$ is a bi-convex function, then for $\Omega \in \mathbb{R},$

$$\left|B_3-\Omega B_2^2\right|\le \left\{\begin{array}{c}1-\Omega for\Omega <{\displaystyle \frac{2}{3}}\\ \\ {\displaystyle \frac{1}{3}}for{\displaystyle \frac{2}{3}}\le \Omega \le {\displaystyle \frac{4}{3}}\\ \\ \Omega -1for\Omega >{\displaystyle \frac{4}{3}}\end{array}\right.$$

(19)

Lemma 3

([25]). If $\Phi \left(\xi \right)=\xi +{\displaystyle \sum _{n\ge 2}}{B}_n{\xi }^n\left(n\in \mathbb{N},\xi \in \mathbb{D}\right)$ is a convex function, then for $\Omega \in \mathbb{R},$

$$\left|B_3-\Omega B_2^2\right|\le \left\{\begin{array}{c}1-\Omega for\Omega <{\displaystyle \frac{2}{3}}\\ \\ 1for{\displaystyle \frac{2}{3}}\le \Omega \le {\displaystyle \frac{4}{3}}\\ \\ \Omega -1for\Omega >{\displaystyle \frac{4}{3}}\end{array}\right..$$

(20)

We introduced new subclasses of bi-univalent functions with bounded boundary rotation for which we derived estimates on the initial coefficients $\left|a_2\right|, \left|a_3\right|$ and $\left|a_4\right|$. Also, we verified the special cases. Additionally, the well-known Fekete–Szegö inequalities were calculated. Our results have several interesting corollaries. Moreover, some of the results extend those of Breaz et al. [26] and improve some inequalities obtained by Srivastava et al. [16].

Unless otherwise stated, $g,\varphi$ and $\psi$ are given in Equations (11), (12), and (13), respectively.

3. Coefficient Bounds Related to ${\mathbb{K}}_{\Sigma }^{\mathbf{\delta }}(\mathit{k},\mathit{\beta })$

Definition 4.

Let $j\in {\mathbb{A}}_{\Sigma }$ given by (1) such that $j^{\prime }\left(\xi \right)\ne 0$ on $\mathbb{D}$. Then j is said to be $\delta -$bi-close-to-convex function with a bounded boundary rotation of order β denoted by ${\mathbb{K}}_{\Sigma }^{\delta }(k,\beta )(0\le \beta <1,$$2\le k\le 4),$ if there exist bi-convex functions $\varphi ,\psi \in {\mathbb{C}}_{\Sigma }$ such that

$$Re\left\{{\displaystyle \frac{{\left[\left(1-\delta \right)j\left(\xi \right)+\delta \xi j^{\prime }\left(\xi \right)\right]}^{\prime }}{{\varphi }^{\prime }\left(\xi \right)}}\right\}\in {\mathbb{P}}_k\left(\beta \right)(\delta \ge 0,\mathit{and}\xi \in \mathbb{D}),$$

(21)

and

$$Re\left\{{\displaystyle \frac{{\left[\left(1-\delta \right)g\left(w\right)+\delta w{g}^{\prime }\left(w\right)\right]}^{\prime }}{{\psi }^{\prime }\left(w\right)}}\right\}\in {\mathbb{P}}_k\left(\beta \right)(\delta \ge 0,\mathit{and}w\in \mathbb{D}),$$

(22)

where the function g is defined by (11).

Remark 1.

(i) 

${\mathbb{K}}_{\Sigma }^0(k,\beta )=$${\mathbb{K}}_{\Sigma }(k,\beta )$$\left(0\le \beta <1\mathrm{and}2\le k\le 4\right)$ is the subset of bi-close-to-convex functions with a bounded boundary rotation of order β, introduced by Breaz et al. [26].

(ii) 

${\mathbb{K}}_{\Sigma }^1(k,\beta )=$$Q{\mathbb{C}}_{\Sigma }(k,\beta )$$\left(0\le \beta <1\mathrm{and}2\le k\le 4\right)$ is the class of bi-quasi-convex functions with a bounded boundary rotation of order β.

Next, we have the initial coefficient bounds and $\left|a_3-\Omega a_2^2\right|$ for the class ${\mathbb{K}}_{\Sigma }^{\delta }(k,\beta )$.

Theorem 1.

Let $j\left(\xi \right),$ in the expansion (1) be in ${\mathbb{K}}_{\Sigma }^{\delta }(k,\beta ),(0\le \beta <1$, $2\le k\le 4).$ Then,

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{1+k\left(1-\beta \right)}{1+2\delta }},}$$

(23)

$$\left|a_3\right|\le {\displaystyle \frac{1}{1+2\delta }}\left[1+k\left(1-\beta \right)\right],$$

(24)

and

$$\left|a_4\right|\le {\displaystyle \frac{1}{1+3\delta }}\left[1+{\displaystyle \frac{3}{2}}k\left(1-\beta \right)\right].$$

(25)

Further, if Ω is real, then

$$\left|a_3-\Omega a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{1}{1+2\delta }}\left[\left(1-\Omega \right)\left(1+\eta \right)\right]\mathit{for}\Omega <0\\ \\ {\displaystyle \frac{1}{3\left(1+2\delta \right)}}\left[\left(1-\Omega \right)\left(3+2\eta \right)+\eta \right]\mathit{for}0\le \Omega <{\displaystyle \frac{2}{3}}\\ \\ {\displaystyle \frac{1}{3\left(1+2\delta \right)}}\left[\left(1+3\eta -2\eta \Omega \right)\right]\mathit{for}{\displaystyle \frac{2}{3}}\le \Omega <1\\ \\ {\displaystyle \frac{1}{3\left(1+2\delta \right)}}\left[\left(1+2\eta \Omega -\eta \right)\right]\mathit{for}1\le \Omega <{\displaystyle \frac{4}{3}}\\ \\ {\displaystyle \frac{1}{3\left(1+2\delta \right)}}\left[\left(\Omega -1\right)\left(3+2\eta \right)+\eta \right]\mathit{for}{\displaystyle \frac{4}{3}}\le \Omega <2\\ \\ {\displaystyle \frac{\Omega -1}{\left(1+2\delta \right)}}\left[\left(1+\eta \right)\right]\mathit{for}\Omega ⩾2\end{array}\right.,$$

(26)

where

$$\eta \le k\left(1-\beta \right).$$

(27)

Proof. 

Let $g,\varphi$, and $\psi$ be expressed as (11), (12), and (13), respectively. Since $j\left(\xi \right)\in {\mathbb{K}}_{\Sigma }^{\delta }(k,\beta ),$ then Definition 4 gives

$${\displaystyle \frac{{\left[\left(1-\delta \right)j\left(\xi \right)+\delta \xi j^{\prime }\left(\xi \right)\right]}^{\prime }}{{\varphi }^{\prime }\left(\xi \right)}}=p\left(\xi \right),p\in {\mathbb{P}}_k\left(\beta \right),$$

(28)

and

$${\displaystyle \frac{{\left[\left(1-\delta \right)g\left(w\right)+\delta w{g}^{\prime }\left(w\right)\right]}^{\prime }}{{\psi }^{\prime }\left(w\right)}}=q\left(w\right),q\in {\mathbb{P}}_k\left(\beta \right),$$

(29)

where

$$p\left(\xi \right)=1+p_1\xi +p_2{\xi }^2+\dots ,\xi \in \mathbb{D},$$

(30)

and

$$q\left(w\right)=1+q_{1}w+q_2{w}^2+\dots ,w\in \mathbb{D}.$$

(31)

By comparing the coefficients in Equation (28) with Equation (30) and the coefficients in Equation (29) with Equation (31), we obtain the following:

$$2\left(1+\delta \right)a_2=2g_2+p_1,$$

(32)

$$3\left(1+2\delta \right)a_3=3g_3+2g_2{p}_1+p_2,$$

(33)

$$-2\left(1+\delta \right)a_2=-2g_2+q_1,$$

(34)

$$3\left(1+2\delta \right)\left(2a_2^2-a_3\right)=3\left(2g_2^2-g_3\right)-2g_2{q}_1+q_2,$$

(35)

and

$$4\left(1+3\delta \right)a_4=4g_4+3p_1{g}_3+2p_2{g}_2+p_3.$$

(36)

From Equations (32) and (34), it is clear that $p_1=-q_1$. Adding Equations (33) and (35) yields the following:

$$6\left(1+2\delta \right)a_2^2=6g_2^2+2g_2\left(p_1-q_1\right)+p_2+q_2.$$

(37)

By using the bound for convex functions, we have $|g_n|\le 1,\forall n\ge 2$. By the relation $p_1=-q_1$, $|g_n|\le 1$, and using Lemma 1, (17), (18) and applying in (37), we arrive at

$${\left|a_2\right|}^2\le {\displaystyle \frac{1+k(1-\beta )}{1+2\delta }}.$$

(38)

This gives (23). Using inequalities $|g_n|\le 1$, $\forall n\ge 2,$ (17) and (18) in (33), we have (24). Next, in order to find the bound on $\left|a_4\right|$, in (25), we employ the same process in relation (36). Now, by (33) and (37), for all $\Omega \in \mathbb{R},$

$$a_3-\Omega a_2^2={\displaystyle \frac{1}{1+2\delta }}\left[\left(g_3-\Omega g_2^2\right)+{\displaystyle \frac{2g_2{p}_1}{3}}\left(1-\Omega \right)+{\displaystyle \frac{p_2}{6}}\left(2-\Omega \right)-{\displaystyle \frac{q_2\Omega }{6}}\right].$$

(39)

By taking the absolute value of both sides of the above equation and employing Lemma 1, (17), (18), and applying $|g_n|\le 1$, $\forall n\ge 2$, we obtain

$$\left|a_3-\Omega a_2^2\right|={\displaystyle \frac{1}{1+2\delta }}\left[\left|g_3-\Omega g_2^2\right|+{\displaystyle \frac{2k(1-\beta )}{3}}\left|1-\Omega \right|+{\displaystyle \frac{k(1-\beta )}{6}}\left(\left|2-\Omega \right|+\left|\Omega \right|\right)\right].$$

(40)

Employing Lemma 2, (26) can be achieved. This concludes the proof. □

Remark 2.

(i) 

In Theorem 1, if we take $\delta =0$, then we have the result of by Breaz et al. ([26], Theorem 1).

(ii) 

Putting $\delta =0$, $1-{\displaystyle \frac{1}{k}}\le \beta <1,$ and $2\le k\le 4$ in Theorem 1, we have a result due to Breaz et al. ([26], Corollary 3).

(iii) 

Putting $\delta =0$ and ${\displaystyle \frac{1}{2}}\le \beta <1$ in Theorem 1, we have a result due to Breaz et al. ([26], Corollary 4).

Corollary 1.

Let $j\in {\mathbb{K}}_{\Sigma }^0(k,0)\equiv {\mathbb{K}}_{\Sigma }\left(k\right)$, $(2\le k\le 4).$ Then,

$$\left|a_2\right|\le \sqrt{1+k,}$$

$$\left|a_3\right|\le 1+k,$$

and

$$\left|a_4\right|\le 1+{\displaystyle \frac{3}{2}}k.$$

Remark 3.

For $\delta =0$ and $\beta =0$, thus Corollary 1 verifies bounds of $\left|a_2\right|$ and $\left|a_3\right|$ and improves $\left|a_4\right|$ obtained in Breaz et al. ([26], Corollary 1).

If $\delta =0$ and $k=2$, then Theorem 1 agrees with the following result proved by Breaz et al. ([26], Corollary 2):

Corollary 2.

Let $j\left(\xi \right)$ in the expansion (1) be in the class ${\mathbb{K}}_{\Sigma }^0(2,\beta )\equiv {\mathbb{K}}_{\Sigma }\left(\beta \right)(0\le \beta <1)$. Then,

$$\left|a_2\right|\le \sqrt{3-2\beta ,}$$

$$\left|a_3\right|\le 3-2\beta ,$$

and

$$\left|a_4\right|\le 4-3\beta .$$

The bounds of $\left|a_2\right|$ and $\left|a_3\right|$ in Corollary 2, verify the result given in [22] for the subclass ${\mathbb{K}}_{\Sigma }(2,\beta )(0\le \beta <1).$

Remark 4.

Choosing $\delta =0$, $k=2$ within Theorem 1, then it verifies the bounds of $\left|a_2\right|$ and $\left|a_3\right|$ and improves those of $\left|a_4\right|$ obtained in Breaz et al. ([26], Corollary 2).

Since every bi-convex function is convex, we can state the following remark.

Remark 5.

Instead of using Lemma 2, if we apply Lemma 3, inequality (26) becomes

$$\left|a_3-\Omega a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{1}{1+2\delta }}\left[\left(1-\Omega \right)\left(1+\eta \right)\right]\mathit{for}\Omega <0\\ \\ {\displaystyle \frac{1}{\left(1+2\delta \right)}}\left[\left(1-\Omega \right)\left(3+2\eta \right)+\eta \right]\mathit{for}0\le \Omega <{\displaystyle \frac{2}{3}}\\ \\ {\displaystyle \frac{1}{\left(1+2\delta \right)}}\left[\left(1+3\eta -2\eta \Omega \right)\right]\mathit{for}{\displaystyle \frac{2}{3}}\le \Omega <1\\ \\ {\displaystyle \frac{1}{\left(1+2\delta \right)}}\left[\left(1+2\eta \Omega -\eta \right)\right]\mathit{for}1\le \Omega <{\displaystyle \frac{4}{3}}\\ \\ {\displaystyle \frac{1}{\left(1+2\delta \right)}}\left[\left(\Omega -1\right)\left(3+2\eta \right)+\eta \right]\mathit{for}{\displaystyle \frac{4}{3}}\le \Omega <2\\ \\ {\displaystyle \frac{\Omega -1}{\left(1+2\delta \right)}}\left[\left(1+\eta \right)\right]\mathit{for}\Omega ⩾2\end{array}\right.$$

(41)

where $\eta \le k\left(1-\beta \right).$

Remark 6.

(i) 

For $\delta =0,$ the inequality (41) coincides with those of Breaz et al. ([26], Theorem 1).

(ii) 

For $\delta =0$, $k=2,$ the inequality (41) reduces to result of Sivasubramanian et al. [22].

Taking $\varphi \left(\xi \right)=\xi ,$ we can obtain the following theorem.

Theorem 2.

Let $j\left(\xi \right),$ in the expansion (1) be in the class ${\mathbb{K}}_{\Sigma }^{\delta }(k,\beta ),(0\le \beta <1,2\le k\le 4).$ Then,

$$\left|a_2\right|\le min\left\{{\displaystyle \frac{k\left(1-\beta \right)}{2\left(1+\delta \right)}},\sqrt{{\displaystyle \frac{k\left(1-\beta \right)}{3\left(1+2\delta \right)}}}\right\},$$

(42)

$$\left|a_3\right|\le {\displaystyle \frac{k\left(1-\beta \right)}{3\left(1+2\delta \right)}},$$

(43)

$$\left|a_4\right|\le {\displaystyle \frac{k\left(1-\beta \right)}{4\left(1+3\delta \right),}}$$

(44)

$$\left|a_3-2a_2^2\right|\le {\displaystyle \frac{k\left(1-\beta \right)}{3\left(1+2\delta \right)}},$$

(45)

and

$$\left|a_3-a_2^2\right|\le {\displaystyle \frac{2k\left(1-\beta \right)}{3\left(1+2\delta \right)}}$$

(46)

Remark 7.

(i) 

Substituting $\delta =0$ into Theorem 2, we arrive at the inequalities of Breaz et al. ([26], Theorem 2).

(ii) 

Substituting $\delta =0$ and $k=2$ into Theorem 2, we arrive at the inequalities of Breaz et al. ([26], Corollary 5).

Remark 8.

(i) 

For $\delta =0$ and $k=2$ in Inequality (43), we obtain

$$\left|a_3\right|\le {\displaystyle \frac{2\left(1-\beta \right)}{3}}<{\displaystyle \frac{\left(1-\beta \right)\left(5-3\beta \right)}{3}},$$

which verifies that the bounds of $\left|a_3\right|$ are less than the corresponding bounds in Srivastava et al. [16].

(ii) 

For $\delta =0$ and $k=2,$ the inequality (46) coincides with those obtained in [24].

4. Particular Coefficient Bounds ${\mathbb{K}}_{\mathbf{2},\Sigma }^{\mathbf{\delta }}(\mathit{k},\mathit{\beta })$

In the recent section, for some choices of the convex function $\varphi \left(\xi \right)$ we derive the coefficients bounds. For $\varphi \left(\xi \right)={\displaystyle \frac{\xi }{1-\xi }}$, we have the subclass ${\mathbb{K}}_{2,\Sigma }^{\delta }(k,\beta )\subset {\mathbb{K}}_{\Sigma }^{\delta }(k,\beta )$. We obtain the corresponding results.

Theorem 3.

Let $j\left(\xi \right)$ given in (1) be in ${\mathbb{K}}_{2,\Sigma }^{\delta }(k,\beta ),(0\le \beta <1,$$2\le k\le 4).$ Then, $\exists \varphi \left(\xi \right)={\displaystyle \frac{\xi }{1-\xi }},$

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{k\left(1-\beta \right)+1}{\left(1+2\delta \right)}},}$$

(47)

$$\left|a_3\right|\le {\displaystyle \frac{k\left(1-\beta \right)+1}{\left(1+2\delta \right)}},$$

(48)

and

$$\left|a_3-\Omega a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{1}{(1+2\delta )}}\left[\left(1+\tau \right)\left(1-\Omega \right)\right]\mathit{for}\Omega <0\\ \\ {\displaystyle \frac{1}{(1+2\delta )}}\left[\tau -\Omega +1-{\displaystyle \frac{2\tau \Omega }{3}}\right]\mathit{for}0\le \Omega <1\\ \\ {\displaystyle \frac{1}{(1+2\delta )}}\left[\Omega -1-{\displaystyle \frac{\tau }{3}}+{\displaystyle \frac{2\tau \Omega }{3}}\right]\mathit{for}1\le \Omega <2\\ \\ {\displaystyle \frac{1}{(1+2\delta )}}\left[\left(1+\tau \right)\left(\Omega -1\right)\right]\mathit{for}\Omega \ge 2\end{array}\right.,$$

(49)

where

$$\tau \le k\left(1-\beta \right).$$

(50)

Proof. 

Suppose that $j\in K_{2,\Sigma }^{\delta }(k,\beta )$; then,

$${\left[\left(1-\delta \right)h\left(\xi \right)+\delta \xi h^{\prime }\left(\xi \right)\right]}^{\prime }=p\left(\xi \right){\varphi }^{\prime }\left(\xi \right)$$

(51)

and

$${\left[\left(1-\delta \right)g\left(w\right)+\delta w{g}^{\prime }\left(w\right)\right]}^{\prime }=q\left(w\right){\psi }^{\prime }\left(w\right).$$

(52)

Since $g=j^{-1}$ is given by Equation (11), $\varphi \left(\xi \right)={\displaystyle \frac{\xi }{1-\xi }}$, $\psi \left(w\right)=w-w^2+w^3-\dots$, and $p\left(\xi \right),q\left(w\right)$ have the forms (15) and (16), respectively. From Equations (51) and (52), we arrive at

$$2(1+\delta )a_2=p_1+2,$$

(53)

$$3(1+2\delta )a_3=p_2+2p_1+3,$$

(54)

$$-2(1+\delta )a_2=q_1-2,$$

(55)

and

$$(6a_2^2-3a_3)(1+2\delta )=q_2-2q_1+3.$$

(56)

From (53) and (55), we have

$$p_1+q_1=0.$$

(57)

Also, from (54), (56), and (57), we find that

$$6a_2^2(1+2\delta )=q_2+p_2+4p_1+6.$$

(58)

Using (17) and (18) in (58), we obtain

$${\left|a_2\right|}^2={\displaystyle \frac{k\left(1-\beta \right)+1}{\left(1+2\delta \right)}}.$$

(59)

This gives (47). Applying (17) and (18) into (54) gives (48).

Now, by (54) and (58), for real $\Omega ,$ we have

$$a_3-\Omega a_2^2={\displaystyle \frac{1}{\left(1+2\delta \right)}}\left[{\displaystyle \frac{p_2\left(2-\Omega \right)}{6}}+\left({\displaystyle \frac{2p_1}{3}}+1\right)\left(1-\Omega \right)-{\displaystyle \frac{q_2\Omega }{6}}\right].$$

(60)

Hence,

$$\left|a_3-\Omega a_2^2\right|\le {\displaystyle \frac{1}{\left(1+2\delta \right)}}\left[{\displaystyle \frac{k(1-\beta )}{6}}\left|2-\Omega \right|+\left({\displaystyle \frac{2k(1-\beta )}{3}}+1\right)\left|1-\Omega \right|+{\displaystyle \frac{k(1-\beta )}{6}}\left|\Omega \right|\right].$$

(61)

By considering different ranges of $\Omega$, we conclude the expression given in (49). This ends the proof. □

Remark 9.

(i) 

Substituting $\delta =0$ into Theorem 3, we arrive at the result previously proven by Breaz et al. ([26], Theorem 4).

(ii) 

Substituting $\delta =0$ and $\beta =0$ into Theorem 3, we arrive at the result previously proven by Breaz et al. ([26], Corollary 8).

(iii) 

Substituting $\delta =0$ and $k=2$ into Theorem 3, we arrive at the results previously proven by Breaz et al. ([26], Corollary 9) and Sivasubramanian et al. [22].

(iv) 

Substituting $\delta =0$, $1-{\displaystyle \frac{1}{k}}\le \beta <1$ and $k\ge 2$ into Theorem 3, we arrive at the result previously proven by Breaz et al. ([26], Corollary 10).

Finally, we select the function $\varphi \left(\xi \right)=-log(1-\xi )$ and denote this class as ${\mathbb{K}}_{\Sigma }^{\delta }(k,\beta )$, which is further denoted as ${\mathbb{K}}_{3,\Sigma }^{\delta }(k,\beta )$. The corresponding results are achieved below.

Theorem 4.

Let j given by (1) and in ${\mathbb{K}}_{3,\Sigma }^{\delta }(k,\beta ),(0\le \beta <1,2\le k\le 4).$ Then, $\exists \varphi \left(\xi \right)=-log(1-\xi )$,

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{8k\left(1-\beta \right)+3}{12\left(1+2\delta \right)}},}$$

(62)

$$\left|a_3\right|\le {\displaystyle \frac{2k\left(1-\beta \right)+1}{3\left(1+2\delta \right)}},$$

(63)

and

$$\left|a_3-4a_2^2\right|\le {\displaystyle \frac{2+6k\left(1-\beta \right)}{3\left(1+2\delta \right)}}.$$

(64)

Proof. 

Suppose that $j\in {\mathbb{K}}_{3,\Sigma }^{\delta }(k,\beta ),$ then

$${\left[\left(1-\delta \right)j\left(\xi \right)+\delta \xi j^{\prime }\left(\xi \right)\right]}^{\prime }=p\left(\xi \right){\varphi }^{\prime }\left(\xi \right),$$

(65)

and

$${\left[\left(1-\delta \right)g\left(w\right)+\delta w{g}^{\prime }\left(w\right)\right]}^{\prime }=q\left(w\right){\psi }^{\prime }\left(w\right).$$

(66)

Since $g=j^{-1}$ is defined by Equation (11), $\varphi \left(\xi \right)=-log(1-\xi )={\displaystyle \sum _{n=1}^{\infty }}$${\displaystyle \frac{{\xi }^n}{n}}$, $\psi \left(w\right)=w-{\displaystyle \frac{1}{2}}{w}^2+{\displaystyle \frac{1}{6}}{w}^3-\dots$, and $p\left(\xi \right),$$q\left(w\right)$ have the forms (15) and (16), respectively.

From Equations (65) and (66), we obtain

$$2(1+\delta )a_2=p_1+1,$$

(67)

$$3(1+2\delta )a_3=p_2+p_1+1,$$

(68)

$$-2(1+\delta )a_2=q_1-1,$$

(69)

and

$$(6a_2^2-3a_3)(1+2\delta )=q_2-q_1+{\displaystyle \frac{1}{2}},$$

(70)

From (67) and (69), we have

$$p_1+q_1=0$$

(71)

Also, from (68), (70), and (71), we find that

$$6a_2^2(1+2\delta )=q_2+p_2+2p_1+{\displaystyle \frac{3}{2}}.$$

(72)

Using (17) and (18) in (72), we obtain

$${\left|a_2\right|}^2\le {\displaystyle \frac{8k\left(1-\beta \right)+3}{12\left(1+2\delta \right)}},$$

(73)

This gives (62). Applying (17) and (18) into (68) gives (63). Now, from (68) and (72), we obtain

$$3a_3-12a_2^2={\displaystyle \frac{-2-2q_2-p_2-3p_1}{\left(1+2\delta \right)}}.$$

(74)

Using (17) and (18) in (74) now gives (64). This finalizes the proof. □

Remark 10.

For $\delta =0$, Theorem 4 agrees with Breaz et al. ([26], Theorem 5).

Corollary 3.

Let j given by (1) and in the class ${\mathbb{K}}_{3,\Sigma }^0(k,\beta )\equiv {\mathbb{K}}_{3,\Sigma }(k,\beta )$$,0\le \beta <1$ and $2\le k\le 4.$ Then, $\exists \varphi \left(\xi \right)=-log(1-\xi ),$

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{8k\left(1-\beta \right)+3}{12}},}$$

$$\left|a_3\right|\le {\displaystyle \frac{2k\left(1-\beta \right)+1}{3}},$$

and

$$\left|a_3-4a_2^2\right|\le {\displaystyle \frac{2+6k\left(1-\beta \right)}{3}}.$$

Remark 11.

For $\delta =0$, Theorem 4 verifies the $\left|a_3\right|$ bound and improves those of $\left|a_2\right|$ and $\left|a_3-4a_2^2\right|$ obtained in Breaz et al. ([26], Theorem 5).

For $\delta =0$ and $\beta =0$, Theorem 4 agrees with the next corollary which was given by Breaz et al. ([26], Corollary 11).

Corollary 4.

Let $j\left(\xi \right),$ in the expansion (1) be in the class ${\mathbb{K}}_{3,\Sigma }^{\delta }(k,0)\equiv {\mathbb{K}}_{3,\Sigma }^{\delta }\left(k\right)$, $2\le k\le 4.$ Then, $\exists \varphi \left(\xi \right)=-log(1-\xi )$,

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{8k+3}{12}},}$$

$$\left|a_3\right|\le {\displaystyle \frac{2k+1}{3}},$$

and

$$\left|a_3-4a_2^2\right|\le {\displaystyle \frac{2+6k}{3}}.$$

Remark 12.

If $\delta =0,\beta =0$, then Theorem 4 confirms the $\left|a_3\right|$ bound and improves the bounds of $\left|a_2\right|$ and $\left|a_3-4a_2^2\right|$ previously derived by Breaz et al. ([26], Corollary 11).

Furthermore, choosing $\delta =0,k=2$ in Theorem 4, we obtain the corresponding bounds established by Breaz et al. ([26], Corollary 12):

Corollary 5.

Let $j\left(\xi \right),$ in the expansion (1) be in the class ${\mathbb{K}}_{3,\Sigma }^0(2,\beta )\equiv {\mathbb{K}}_{3,\Sigma }(2,\beta )$$,0\le \beta <1$ and $2\le k\le 4.$ Then, $\exists \varphi \left(\xi \right)=-log(1-\xi )$, such that

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{16\left(1-\beta \right)+3}{12}},}$$

$$\left|a_3\right|\le {\displaystyle \frac{4\left(1-\beta \right)+1}{3}},$$

and

$$\left|a_3-4a_2^2\right|\le {\displaystyle \frac{2+12\left(1-\beta \right)}{3}}.$$

Remark 13.

If $\delta =0,$$k=2$, then Theorem 4 confirms the $\left|a_3\right|$ bound and improves bounds of $\left|a_2\right|$ and $\left|a_3-4a_2^2\right|$ previously obtained by Breaz et al. ([26], Corollary 12).

If $\delta =1$, then ${\mathbb{K}}_{\Sigma }^1(k,\beta )=Q{\mathbb{C}}_{\Sigma }(k,\beta )$, the set of all bi-quasi-convex functions with bounded boundary rotation of order $\beta$.

Theorem 5.

Let $j\left(\xi \right),$ in the expansion (1) be in the class $Q{\mathbb{C}}_{\Sigma }(k,\beta ),$$0\le \beta <1$ and $2\le k\le 4.$ Then,

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{1+k\left(1-\beta \right)}{3}},}$$

(75)

$$\left|a_3\right|\le {\displaystyle \frac{1}{3}}\left[1+k\left(1-\beta \right)\right],$$

(76)

and

$$\left|a_4\right|\le {\displaystyle \frac{1}{4}}\left[1+{\displaystyle \frac{3}{2}}k\left(1-\beta \right)\right].$$

(77)

Further, if Ω is real, then

$$\left|a_3-\Omega a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{1}{3}}\left[\left(1-\Omega \right)\left(1+\eta \right)\right]\mathit{for}\Omega <0\\ \\ {\displaystyle \frac{1}{9}}\left[\left(1-\Omega \right)\left(3+2\eta \right)+\eta \right]\mathit{for}0\le \Omega <{\displaystyle \frac{2}{3}}\\ \\ {\displaystyle \frac{1}{9}}\left[\left(1+3\eta -2\eta \Omega \right)\right]\mathit{for}{\displaystyle \frac{2}{3}}\le \Omega <1\\ \\ {\displaystyle \frac{1}{9}}\left[\left(1+2\eta \Omega -\eta \right)\right]\mathit{for}1\le \Omega <{\displaystyle \frac{4}{3}}\\ \\ {\displaystyle \frac{1}{9}}\left[\left(\Omega -1\right)\left(3+2\eta \right)+\eta \right]\mathit{for}{\displaystyle \frac{4}{3}}\le \Omega <2\\ \\ {\displaystyle \frac{1}{3}}\left[\left(\Omega -1\right)\left(1+\eta \right)\right]\mathit{for}\Omega ⩾2\end{array}\right.$$

(78)

where

$$\eta \le k\left(1-\beta \right).$$

(79)

5. Coefficient Bounds Related to ${\mathbb{M}}_{\Sigma }^{\mathbf{\rho }}(\mathit{\alpha },\mathit{k},\mathit{\beta })$

We begin this section by providing the definition of a $\rho -$bi-convex function with a bounded boundary rotation of order $\beta$.

Definition 5.

A function $j\in {\mathbb{A}}_{\Sigma },$ given by (1) is called ρ-bi-convex functions with bounded boundary rotation of order β, denoted by $M_{\Sigma }^{\rho }(\alpha ,k,\beta )\left(0\le \beta <1,2\le k\le 4\right),$ if

$${\left({\displaystyle \frac{\xi j^{\prime }\left(\xi \right)}{j\left(\xi \right)}}\right)}^{\alpha }{\left(1+{\displaystyle \frac{\xi j^{\prime \prime }\left(\xi \right)}{j^{\prime }\left(\xi \right)}}\right)}^{\rho }\in {\mathbb{P}}_k\left(\beta \right)(\alpha ,\rho \in \mathbb{R},\mathrm{and}\xi \in \mathbb{D}),$$

(80)

and

$${\left({\displaystyle \frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}}\right)}^{\alpha }{\left(1+{\displaystyle \frac{w{g}^{\prime \prime }\left(w\right)}{g^{\prime }\left(w\right)}}\right)}^{\rho }\in {\mathbb{P}}_k\left(\beta \right)(\alpha ,\rho \in \mathbb{R},\mathrm{and}w\in \mathbb{D}),$$

(81)

where the function g is defined by (11) and $j^{\prime }\left(\xi \right)\ne 0$ on $\mathbb{D}$.

Remark 14.

(i) 

When $\alpha =1$ and $\rho =0,$${\mathbb{M}}_{\Sigma }^{\rho }(\alpha ,k,\beta )$$\equiv {\mathbb{M}}_{\Sigma }^0(1,k,\beta )\equiv$${\mathbb{S}}_{\Sigma }^{*}(k,\beta )$ (see [11]), this represents the class of bi-starlike functions with bounded boundary rotation of order $\beta .$

(ii) 

When $\alpha =1,$$\rho =0$ and $k=2,$${\mathbb{M}}_{\Sigma }^{\rho }(\alpha ,k,\beta )$$\equiv {\mathbb{M}}_{\Sigma }^0(1,2,\beta )$≡${\mathbb{S}}_{\Sigma }^{*}(2,\beta )\equiv$${\mathbb{S}}_{\Sigma }^{*}\left(\beta \right).$ This denotes the class of bi-starlike functions of order $\beta .$

(iii) 

When $\alpha =0$ and $\rho =1,$${\mathbb{M}}_{\Sigma }^{\rho }(\alpha ,k,\beta )$$\equiv {\mathbb{M}}_{\Sigma }^1(0,k,\beta )\equiv$${\mathbb{C}}_{\Sigma }(k,\beta )$ (see [11]). This represents the class of bi-convex functions with bounded boundary rotation of order $\beta .$

(iv) 

When $\alpha =0$, $\rho =1$, and $k=2,$${\mathbb{M}}_{\Sigma }^{\rho }(\alpha ,k,\beta )$$\equiv {\mathbb{M}}_{\Sigma }^1(0,2,\beta )$≡${\mathbb{C}}_{\Sigma }(2,\beta )\equiv$${\mathbb{C}}_{\Sigma }\left(\beta \right)$. This denotes the class of bi-convex functions of order $\beta .$

Theorem 6.

Let j given by (1) and in ${\mathbb{M}}_{\Sigma }^{\rho }(\alpha ,k,\beta ),$$(0\le \beta <1,2\le k\le 4)$; then,

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{k\left(1-\beta \right)}{2\rho (\alpha +\rho )+\frac{1}{2}\alpha (\alpha +1)}}},\alpha ,\rho \in \mathbb{R}$$

(82)

$$\left|a_3\right|\le {\displaystyle \frac{k\left(1-\beta \right)}{2\rho (\alpha +\rho )+\frac{1}{2}\alpha (\alpha +1)}},$$

(83)

$$\left|a_3-\tau a_2^2\right|\le {\displaystyle \frac{k\left(1-\beta \right)}{2(\alpha +3\rho )}},$$

(84)

$$\left|a_3-\epsilon a_2^2\right|\le {\displaystyle \frac{k\left(1-\beta \right)}{2(\alpha +3\rho )}},$$

(85)

where

$$\tau ={\displaystyle \frac{2\rho (\alpha +\rho +3)+\frac{\alpha }{2}(\alpha +5)}{2(\alpha +3\rho )}}\mathit{and}\epsilon ={\displaystyle \frac{2\rho (3-\alpha -\rho )+\frac{\alpha }{2}(3-\alpha )}{2(\alpha +3\rho )}}.$$

Proof. 

It follows from (80) and (81) that there exist $p,q\in {\mathbb{P}}_k\left(\beta \right)$ such that

$${\left({\displaystyle \frac{\xi j^{\prime }\left(\xi \right)}{j\left(\xi \right)}}\right)}^{\alpha }{\left(1+{\displaystyle \frac{\xi j^{\prime \prime }\left(\xi \right)}{j^{\prime }\left(\xi \right)}}\right)}^{\rho }=p\left(\xi \right)$$

(86)

and

$${\left({\displaystyle \frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}}\right)}^{\alpha }{\left(1+{\displaystyle \frac{w{g}^{\prime \prime }\left(w\right)}{g^{\prime }\left(w\right)}}\right)}^{\rho }=q\left(w\right).$$

(87)

where $g=j^{-1},$$p\left(\xi \right)$, and $q\left(w\right)$ have the forms (11), (15), and (16), respectively.

Now, (86) and (87) are rewritten as

$$1+\left(\alpha +2\rho \right)a_2\xi +\left[2\left(\alpha +3\rho \right)a_3+\left(2\rho \left(\alpha +\rho -3\right)+{\displaystyle \frac{\alpha (\alpha -3)}{2}}\right)a_2^2\right]{\xi }^2+\dots$$

$$=1+p_1\xi +p_2{\xi }^n+\dots$$

(88)

and

$$1-\left(\alpha +2\rho \right)a_{2}w+\left[-2\left(\alpha +3\rho \right)a_3+\left(2\rho \left(\alpha +\rho +3\right)+{\displaystyle \frac{\alpha (\alpha +5)}{2}}\right)a_2^2\right]w^2+\dots$$

$$=1+q_{1}w+q_2{w}^2+\dots ,$$

(89)

respectively. Now, comparing the coefficients of Equations (88) and (89), we obtain

$$\left(\alpha +2\rho \right)a_2=p_1,$$

(90)

$$2\left(\alpha +3\rho \right)a_3+\left(2\rho \left(\alpha +\rho -3\right)+{\displaystyle \frac{\alpha (\alpha -3)}{2}}\right)a_2^2=p_{2,}$$

(91)

$$-\left(\alpha +2\rho \right)a_2=q_1,$$

(92)

and

$$-2\left(\alpha +3\rho \right)a_3+\left(2\rho \left(\alpha +\rho +3\right)+{\displaystyle \frac{\alpha (\alpha +5)}{2}}\right)a_2^2=q_2.$$

(93)

From (90) and (92), we obtain

$$p_1=-q_1.$$

(94)

Adding (91) and (93) yields

$$\left[4\rho (\alpha +\rho )+\alpha (\alpha +1)\right]a_2^2=p_{2+}{q}_2.$$

(95)

Now, using inequalities (17) and (18) in (95), we have

$${\left|a_2\right|}^2\le {\displaystyle \frac{k(1-\beta )}{2\rho (\alpha +\rho )+\frac{\alpha (\alpha +1)}{2}}}.$$

(96)

Thus, we obtain the coefficient of $\left|a_2\right|$ as maintained in (82). Applying (96), (17) and (18) in (91) at once gives (83). Now, (91) is given as

$$a_3-{\displaystyle \frac{\left(2\rho \left(3-\alpha -\rho \right)+\frac{\alpha (3-\alpha )}{2}\right)}{2(\alpha +3\rho )}}{a}_2^2={\displaystyle \frac{p_2}{2(\alpha +3\rho )}}.$$

(97)

Furthermore,

$$\left|a_3-\epsilon a_2^2\right|={\displaystyle \frac{\left|p_2\right|}{2(\alpha +3\rho )}}\le {\displaystyle \frac{k(1-\beta )}{2(\alpha +3\rho )}},$$

(98)

where

$$\epsilon ={\displaystyle \frac{2\rho (3-\alpha -\rho )+\frac{\alpha }{2}(3-\alpha )}{2(\alpha +3\rho )}}.$$

(99)

Now, (93) becomes

$$a_3-{\displaystyle \frac{\left(2\rho \left(3+\alpha +\rho \right)+\frac{\alpha (\alpha +5)}{2}\right)}{2(\alpha +3\rho )}}{a}_2^2={\displaystyle \frac{-q_2}{2(\alpha +3\rho )}}.$$

(100)

Furthermore,

$$\left|a_3-\tau a_2^2\right|={\displaystyle \frac{\left|q_2\right|}{2(\alpha +3\rho )}}\le {\displaystyle \frac{k(1-\beta )}{2(\alpha +3\rho )}},$$

(101)

where

$$\tau ={\displaystyle \frac{\left(2\rho \left(3+\alpha +\rho \right)+\frac{\alpha (\alpha +5)}{2}\right)}{2(\alpha +3\rho )}}.$$

(102)

And the proof is now completed. □

Remark 15.

(i) 

Substituting $\alpha =1,\rho =0$ into Theorem 6, we have the inequalities in Sharma et al. ([27], Corollary 6).

(ii) 

Substituting $\alpha =1$, $\rho =0$ and $k=2$ in Theorem 6, we obtain the result obtained by Mishra and Soren [28].

(iii) 

Substituting $\alpha =1$, $\rho =0,$$\beta =0$ and $k=2$ in Theorem 6, we arrive at the result obtained by Sharma et al. ([27], Corollary 8).

(iv) 

When $\alpha =0$ and $\rho =1$, Theorem 6 agrees with the results Sharma et al. ([27], Corollary 7).

Corollary 6.

Let j in the expansion (1) be in the class ${\mathbb{M}}_{\Sigma }^1(0,k,\beta )\equiv {\mathbb{C}}_{\Sigma }(k,\beta ),$$(0\le \beta <1,$$2\le k\le 4)$; then,

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{k\left(1-\beta \right)}{2}}},$$

$$\left|a_3\right|\le {\displaystyle \frac{k\left(1-\beta \right)}{2}},$$

$$\left|a_3-{\displaystyle \frac{4}{3}}{a}_2^2\right|\le {\displaystyle \frac{k\left(1-\beta \right)}{6}}$$

and

$$\left|a_3-{\displaystyle \frac{2}{3}}{a}_2^2\right|\le {\displaystyle \frac{k\left(1-\beta \right)}{6}}.$$

Remark 16.

For $\alpha =0$ and $\rho =1$, Theorem 6 verifies the coefficient bounds of $\left|a_2\right|$ and $\left|a_3\right|$ and improves the bounds of $\left|a_3-{\displaystyle \frac{1}{2}}{a}_2^2\right|$ and $\left|a_3-{\displaystyle \frac{3}{2}}{a}_2^2\right|,$ obtained by Sharma et al. ([27], Corollary 7), for the family of all bi-convex functions with a bounded boundary rotation of order β, denoted by ${\mathbb{C}}_{\Sigma }(k,\beta )$.

6. Conclusions

This work introduces new subclasses of bi-univalent functions with bounded boundary rotation. $\left|a_2\right|$, $\left|a_3\right|$, and $\left|a_4\right|$ were the initial coefficients for which we obtained estimates. Additionally, we have confirmed the particular circumstances that fit the well-known Brannan and Clunie hypothesis. Furthermore, we have derived the well-known Fekete–Szegö inequality for the recently discovered subclasses of bi-univalent functions. Our findings not only enhance but also expand upon a number of previous results as specific examples.